Is it true that it is not the responsibility of the child's father (the fornicator) to raise and care for the child, and that the burden falls on the mother, and is there any evidence for that?

All of Allah's rulings are just, and a Muslim's reference for legal rulings is the Qur'an and Sunnah (definition). It is understood from the hadith (definition) of Ma'iz and Al-Ghamidiyyah that the person most entitled to the custody of a child born of zina (unlawful sexual intercourse) is its mother. And it is from Allah's mercy that the child is not burdened with the disgrace of its parents' transgression. The mother is not legally obligated towards her child from zina for financial support, breastfeeding, or custody, unless its perishment is feared. "The child is for the bed, and for the fornicator is the stone," meaning that the fornicator has no right to the child, and the lineage of a child born of zina is not established from the fornicating cohabitant. It is attributed to the fornicating woman who gave birth to it, if her giving birth to him is proven. If the fornicator marries the woman with whom he committed zina while she is pregnant with his child, and his paternity is established, then this is ruled upon in worldly judgments.